This thesis deals with stochastic dynamical systems in discrete and continuous
time. Traditionally dynamical systems in continuous time are modelled using
Ordinary Differential Equations (ODEs). Even the most complex system of
ODEs will not be able to capture every detail of a complex system like a natural
ecosystem, and hence residual variation between the model and observations
will always remain. In stochastic state-space models the residual variation is
separated into observation and system noise and a main theme of the thesis
is a proper description of the system noise. Additive Gaussian noise is the
standard approach to introduce system noise, but this may lead to undesirable
consequences for the state variables. In biological models, where the statespace
generally contains positive real numbers only, modelling in the log-domain
ensures positive state variables, however, this transformation is likely to conflict
with the concept of mass balances. One of the central conclusions of the thesis
is that the stochastic formulations should be an integral part of the model
formulation.
As discrete-time stochastic processes are simpler to handle numerically than
continuous-time stochastic processes, I start by considering discrete-time processes.
An novel approach combining multiplicative and additive log-normal
noise has been developed in discrete time, and used to demonstrate the effect of
stochastic forcing in simple discrete-time regime shift models. An approximate
maximum likelihood estimation procedure based on the second order moment
representation of the multiplicative and additive log-normal noise model was
developed and tested in simulation studies.
The transition to continuous-time stochastic models (here Stochastic Differential
Equations (SDEs)) offers the opportunity of embedding parts of the ODE
processes into the stochastic part of the model (the diffusion term). The estimation
method we use here (maximum likelihood and the Extended Kalman
Filter (EKF)) rely on state-independent diffusion, but for a wide class of SDEs
there exist an alternative description (given by the Lamperti transform) of the
input-output relation, where the diffusion term is independent of the state. This
alternative description is used to develop better parametric descriptions of the
diffusion term, while maintaining the opportunity of estimation by standard
software.
Additionally, the state-space formulation facilitates estimation of unobserved
states. Based on estimation of random walk hidden states and examination of
simulated distributions and stationarity characteristics, a methodological framework
for structural identification based on information embedded in the observations
of the system has been developed. The applicability of the methodology
is demonstrated using phytoplankton and nitrogen data from a Danish estuary
as well as bacterial growth data from a controlled experiment.
In summary, the novelty of the work presented here is the introduction of more
appropriate stochastic descriptions in non-linear state-space models, which can
include combinations of additive and multiplicative noise components under various
distributional assumptions. A model identification and estimation framework
for working with such models has been developed and tested using data
from biological and ecological systems typically characterised by non-linear and
non-Gaussian responses.